12 research outputs found
There are no magnetically charged particle-like solutions of the Einstein Yang-Mills equations for Abelian models
We prove that there are no magnetically charged particle-like solutions for
Abelian models in Einstein Yang-Mills, but for non-Abelian models the
possibility remains open. An analysis of the Lie algebraic structure of the
Yang-Mills fields is essential to our results. In one key step of our analysis
we use invariant polynomials to determine which orbits of the gauge group
contain the possible asymptotic Yang-Mills field configurations. Together with
a new horizontal/vertical space decomposition of the Yang-Mills fields this
enables us to overcome some obstacles and complete a dynamical system existence
theorem for asymptotic solutions with nonzero total magnetic charge. We then
prove that these solutions cannot be extended globally for Abelian models and
begin an investigation of the details for non-Abelian models.Comment: 48 pages, 1 figur
Existence of families of spacetimes with a Newtonian limit
J\"urgen Ehlers developed \emph{frame theory} to better understand the
relationship between general relativity and Newtonian gravity. Frame theory
contains a parameter , which can be thought of as , where
is the speed of light. By construction, frame theory is equivalent to general
relativity for , and reduces to Newtonian gravity for .
Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to
study the Newtonian limit \ep \searrow 0 (i.e. ). A number of
ideas relating to frame theory that were introduced by J\"urgen have
subsequently found important applications to the rigorous study of both the
Newtonian limit and post-Newtonian expansions. In this article, we review frame
theory and discuss, in a non-technical fashion, some of the rigorous results on
the Newtonian limit and post-Newtonian expansions that have followed from
J\"urgen's work
Cosmological post-Newtonian expansions to arbitrary order
We prove the existence of a large class of one parameter families of
solutions to the Einstein-Euler equations that depend on the singular parameter
\ep=v_T/c (0<\ep < \ep_0), where is the speed of light, and is a
typical speed of the gravitating fluid. These solutions are shown to exist on a
common spacetime slab M\cong [0,T)\times \Tbb^3, and converge as \ep
\searrow 0 to a solution of the cosmological Poisson-Euler equations of
Newtonian gravity. Moreover, we establish that these solutions can be expanded
in the parameter \ep to any specified order with expansion coefficients that
satisfy \ep-independent (nonlocal) symmetric hyperbolic equations
On the positive mass theorem for manifolds with corners
We study the positive mass theorem for certain non-smooth metrics following
P. Miao's work. Our approach is to smooth the metric using the Ricci flow. As
well as improving some previous results on the behaviour of the ADM mass under
the Ricci flow, we extend the analysis of the zero mass case to higher
dimensions.Comment: 21 pages, incorporated referee's comment
Post-Newtonian expansions for perfect fluids
We prove the existence of a large class of dynamical solutions to the
Einstein-Euler equations that have a first post-Newtonian expansion. The
results here are based on the elliptic-hyperbolic formulation of the
Einstein-Euler equations used in \cite{Oli06}, which contains a singular
parameter \ep = v_T/c, where is a characteristic velocity associated
with the fluid and is the speed of light. As in \cite{Oli06}, energy
estimates on weighted Sobolev spaces are used to analyze the behavior of
solutions to the Einstein-Euler equations in the limit \ep\searrow 0, and to
demonstrate the validity of the first post-Newtonian expansion as an
approximation
Stability of complex hyperbolic space under curvature-normalized Ricci flow
Using the maximal regularity theory for quasilinear parabolic systems, we
prove two stability results of complex hyperbolic space under the
curvature-normalized Ricci flow in complex dimensions two and higher. The first
result is on a closed manifold. The second result is on a complete noncompact
manifold. To prove both results, we fully analyze the structure of the
Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result,
we also define suitably weighted little H\"{o}lder spaces on a complete
noncompact manifold and establish their interpolation properties.Comment: Some typos in version 2 are correcte
Classical Yang-Mills Black hole hair in anti-de Sitter space
The properties of hairy black holes in Einstein–Yang–Mills (EYM) theory are reviewed, focusing on spherically symmetric solutions. In particular, in asymptotically anti-de Sitter space (adS) stable black hole hair is known to exist for frak su(2) EYM. We review recent work in which it is shown that stable hair also exists in frak su(N) EYM for arbitrary N, so that there is no upper limit on how much stable hair a black hole in adS can possess